There has been a discussion about PI-Indexing and irrational number theory here.
Now suppose you want to use indexes of PI to store information (file storage). To get around the fact that the indexes will be bigger than the files you're storing, you use tree of indexes - PI at a certain value is a pointer to PI at a another value.
Here the author writes:
By the pigeonhole principle, no matter how fast you can calculate pi, you cannot actually use this to compress data. The index to relevant sequence is on average >= the size of the data to be stored.
Now I'm looking at the pigeonhole principles - and I'm not connecting to why it says this is not possible. (It sounds plausible).
My question is: Why does the pigegonhole principle tell us that PI-indexing is not possible?
EDIT: A related example - here is an example of a filesystem that stores files as locations in PI.